On The Well-posedness And Asymptotic Behavior Of Several Kinds Of Kinetic Equations

This paper deals with the well-posedness and asymptotic behavior of several kinds of kinetic equations.On one hand,we investigate some collisionless kinetic equations from astrodynamics and plasma-the Vlasov type,such as(Relativistic)Vlasov-Maxwell system;Vlasov-Poisson system;Vlasov-Poisson system with point charges.On the other hand,we are concerned with a mathematical model for flocking and coordinated control,called Cucker-Smale-Fokker-Planck(CS-FP)system.In Chapter 1,we mainly introduce the physical background,research technique and status of researches on the above system,and then abstract the relevant issues.Moreover the contribution of this paper is also presented in this chapter.In Chapter 2,we consider the three-dimensional Vlasov-Poisson system with point charges.By means of constructing a new kinds of microscopic energy functional and macro-scopic energy functional,we prove global existence and uniqueness of a classical solution possibly having infinite kinetic energy specified in Theorem 2.1.1,following from classical Lagrange method.Moreover,the large time behavior in terms of diameters of its velocity-spatial supports is improved to sub-linear estimate specified in Theorem 2.1.2.Based on the results in Chapter 2,for the three-dimensional Vlasov-Poisson system with single point,global existence and polynomial propagation of the high-order "velocity-spatial moments" of weak solutions are also established in Chapter 3,specified in Theorem 3.1.1,following from classical Euler method.In Chapter 4,it is shown that solutions of the(relativistic)Vlasov-Maxwell system with small data converge pointwise to solutions of the Vlasov-Poisson system globally in time at the asymptotic rate of cl-1,as the light speed c tends to infinity,following from the accurate estimate of the electric field and magnetic field defined by the Cauchy problem of(relativistic)Vlasov-Maxwell system,which have been specified in Theorem 4.1.1 and Theorem 4.1.2.In Chapter 5,the Cucker-Smale-Fokker-Planck system is investigated in Rd.For the singular communication weight ?(x)in space,?(x)= O(1/|x|?)with 0<?<lifd=1,0<?<2if d?2,we consider the well-posedness of the Cucker-Smale-Fokker-Planck system with finite ki-netic energy,which is specified in Theorem 5.1.1.In addition,if ? satisfies 0<?<d and the k-th and above order moment in velocity of initial data is bounded,we consider local weak solutions to the Cauchy problem for Cucker-Smale-Fokker-Planck system and obtain a propagation result for any velocity moment of order k and above,which are specified in Theorem 5.1.2.In the end,we discuss some problems for further study about the above system.